To: Honors Calculus Students. From: Mrs. Hausman. Re: Summer work

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1 To: Honors Calculus Students From: Mrs. Hausman Re: Summer work Directions: All students taking Honors Calculus next school year should complete all problems contained in the packet. This material will not be covered in class. You are expected to know all the material enclosed before starting Calculus. During the summer you can me to ask any questions about the material contained in the packet. semi-regularly and will respond when I I will check my can. Don t procrastinate and wait until the last week before school starts to begin completing the packet. My address is mhausmanjwschoois.net. Students should expect to spend approximately 10 hours on the assignments. The packet should be turned in to your instructor on the first day of school, and a grade will be given based on the completeness and accuracy of solutions. You must show your work for problems on a separate sheet of paper. You should be prepared to take a test on the material covered in this packet when you return to school. Have a great summer!

2 5x220x--x Summer Review Packet for Students Entering Calculus Name Complex Fractions When simplifying complex fractions, multiply by a fraction equal to 1 which has a numerator and denominator composed of the common denominator of all the denominators in the complex fraction. Example: 76 6 x-fl x+l x+l 5 x-*l 7x76 5 7x x 2 3x x(x 4) 51 51,t(x4) x4 x4 2(x 4) 3x(x) 5(x)(x4)l(x) 2x 8 + 3x2 3x2 2x + 8 5x221x Simplify each of the foflowing. 25 a a 5+a f D + x x x3 x x, x+lx I I x 2x 3x4 32 x + 3x 4

3 Functions o evaluate a function for a given value, simply plug the value into the function for x. Recall: (j g)(x) = f(g(x)) OR flg(x)] read fofg ofx Means to plug the inside function (in this case in for x in the outside function (in this case, f(x)). xamplc: Given f(x) =2x 2 + I and g(x) = x 4 find f(g(x)). f(g(x))f(x4) =2(x4) 2+l =2(xZ_8x+16)+l f(g(x)) = 2x 216x+32+1 =2x 2 I 6x +33 Let 1(x) = 2x + 1 and g(x) = 2x I. 2 Find each. 6. f(2)= 7. g(3)=_ 8. f(i+l)= 9. J[g(-2)]= 10. g[f(m+2)]= 11 f(x+h)-i(xl Let f(x) = sinx Find each exactly. 12. i(!!.j= 3 f(jz Let f(x)=x 2, g(x)=2x+5, and h(x)=x 2 I. Find each. 14. h[f(_2)] 15. f[g(x-i)]= 16. g[h(x )]= 3

4 Find f(x + f(x) for the given functionf 17. f(x)=9x f(x)52x Intercepts and Points of Intersection To find the x-intercepts, let y = 0 in your equation and solve. To find the y-intercepts, let x =0 in your equation and solve. Example: y=x xint. 22x3 (Lety=O) 0=x 22x3 0=(x3)(x±l) x=iorx3 x intercepts (1,0) and (3,0) yint.(letx=0) y=0 22(0)3 3 y= ylntercept (0,3) Find the x and y intercepts for each. 19. v=2x x2 y=x 21. y=xl6_x2 3-4x 22. y2=x 4

5 Use substitution or elimination method to solve the system of equations. Example x +yl6x390 29=0 x2y Elimination Method Substitution Method x x2 8x F 30 = = 0 (x 3)(x 5) Q x = 3 and x = 5 Plug x=3 and x = 5 into one original y 32y 2 y 29=0 = 0 16 = y2 y=0 y=±4 Points of Intersection (5,4), (5,4) and (3,0) Solve one equation for one variable. y2 = x 2 +16x 39 (1st equation solved for y) x 39)9=0 Plug what y2 is equal x2 (x 2x 2l6x+30=0 x28x+150 (x3)(x5)=0 x = 3 or x 5 to into second equation. (The rest is the same as previous example) Find the point(s) of intersection of the graphs for the given equations. 23. x + y = 8 4xy=7 24. x2+y=6 x + y =4 25 x24y 2--20x64y--172=O 16x 2+4y 2320x+64y+1600=O Interval Notation 26. Complete the table with the appropriate notation or graph. Solution Interval Notation Graph -2< x4 [1,7) 8 5

6 Solve each equation. State your answer in BOTH interval notation and graphically x x--3< Domain and Range Find the domain and range of each function. Write your answer in INTERVAL notation. 30. f(x)=x f(x)= 32. f(x)=3sinx 33. f(x)= xl In verses To find the inverse of a function, simply switch the x and they and solve for the new y value. Example: f(x) = Rewrite f(x) as y y1 Switchxandy x = Solve for your new y (x) 3 =(ji) Cube both sides x3=y+l Simplify y=x 3l Solvefory f (x) = I Find the inverse for each function. Rewrite in inverse notation 34. f(x) = 2x f(x) 6

7 Also, recall that to PROVE one function is an inverse of another function, you need to show that: f(g(x))- g(f(x))=x Example: j 2 If: f(x) = and g(x) = 4x + 9 showfl x) andg(x) are inverses of each other. g(f(x))= (4x+9)_9 =_t 4x x 4 f(g(x)) = g(f(x)) = x therefore they are inverses of each other. Provefand are inverses of each other. 36. f(x) L g(x) = 37. f(x) =9- x, x 0

8 Equation of a line Slope intercept form: y = mx - Point-slope form: y b Vertical line: x = c (slope is undefined) y1 = m(x x1) Horizontal line: y = c (slope is 0) 38. Use slope-intercept form to find the equation of the line having a slope of 3 and a y-intercept of 5, 39. Determine the equation of a line passing through the point (5, -3) with an undefined slope. 40. Determine the equation of a line passing through the point (-4, 2) with a slope of Use point-slope fbrm to find the equation of the line passing through the point (0, 5) with a slope of 2/ Find the equation of a line passing through the point (2, 8) and parallel to the line y =x Find the equation of a line perpendicular to the y- axis passing through the point (4, 7). 44. Find the equation of a line passing through the points (-3, 6) and (1, 2). 45. Find the equation of a line with an x-intercept (2, 0) and a y-intercept (0, 3). 8

9 Solve each equation or inequality. 46. x+41 3= <3(x--2)< / 7 + 3x = log x + log (x + 3) = =q+q 9

10 Trig Questions: 52. Find 9ftanO=1andOO22t 53. sin2s+cos 2O 54. the period of y 2 sin 5x 55. arcsin (2) 56. tan

11 Log Questions: 57. a. log 8 (x + 1) log 8 x = log 8 4 b. log, 27 = x 58. a. log 5 b. logx=-log64+logl2l Radian and Degree Measure I8O rradians 1 Use to get rid of radians and Use to get rid of degrees and I ir radians l80 Lconve to degrees, convert to radians. 59. Convert to degrees: a. b. c radians Convert to radians: a. 45 b. IT c

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